Magic Squares for Special Occasions

Age 11 to 16
Article by Charlie Gilderdale

Published October 2000,February 2011.


In this article, Charlie Gilderdale records a meeting with the famous maths teach P.K. Srinivasan, who sadly passed away in 2005. You can read more about Srinivasan in his Wikipedia entry

On a recent visit to India I was fortunate to meet P.K.Srinivasan. He welcomed us on the 15th August 2000 with a Magic Square which contained the date on the first row:

15 8 20 00
14 11 13 5
2 18 1 22
12 6 9 16

He explained how this can be done for any date:

I will use this grid for reference purposes:

a b c d
e f g h
i j k l
m n o p


(i) Place the special date
in the first row
eg. Christmas Day 2000

25 12 20 00
       
       
       
(ii) b+c = m+p [there are many different possible values]


25 12 20 00
       
       
14     18
 
(iii) a+p = g+j [there are many different possible values]

25 12 20 00
    10  
  33    
14     18
(iv) m+d = f+k [there are many different possible values]

25 12 20 00
  3 10  
  33 11  
14     18
 
(v) b+n = g+k
[therefore n = 9]

25 12 20 00
  3 10  
  33 11  
14 9   18
   
(vi) c+o = f+j
[therefore o = 16]

25 12 20 00
  3 10  
  33 11  
14 9 16 18
(check: a+d = n+o)
 
(vii) a+m = h+l
[there are many different
possible values]

25 12 20 00
  3 10 31
  33 11 8
14 9 16 18
   
(viii) All rows, columns and diagonals must add up to the same total, so e = 13 and i = 5.

25 12 20 00
13 3 10 31
5 33 11 8
14 9 16 18
(check: d+p = e+i)

There are many different solutions, and the problem is trivial if we are allowed to repeat numbers; so the challenge is to complete the square without using any number more than once (but you will need to use negative numbers if the numbers in the top row add to less than 34).

Can you complete the Christmas Day Magic Square in a different way?

Can you complete a magic square with the date of your birthday in the top row?

There are some articles about magic squares on the NRICH website which you may like to see, Magic Squares, its follow-up Magic Squares II and also Magic Sums and Products . A computer program to find magic squares shows how to program a computer to follow the method for finding magic squares described in this article. If this has whet your appetite there are some problems in the Archive which you might like to have a go at tackling (you can use the search box in the left hand margin to find them).

P.K.Srinivasan was the Curator-Director of the Ramanujan Museum and Maths Education Centre in Chennai. 


A very young-looking Charlie Gilderdale and P.K.Srinivasan working on some mathematics together.